Schwarzschild metric in Kruskal coord's

[reverbtank]

Homework Statement

This question is very simple, but it is driving me mad.

Show that the Schwarzschild metric in Kruskal coordinates takes the form

ds² = (32M³/r)e^-r/2M(-dv²+du²) +r²(d(theta)² + sin²(theta)*d(phi)²)

Homework Equations

The equations are just those defining the Kruskal coordinates
(1) u² - v² = (r/2M -1)e^r/2M

and
(2) v/u = tanh(t/4M)

The Attempt at a Solution

I've tried this numerous times without luck. Usually I start by solving the second eqt'n for t and then take its total time derivative. This gets me dt² to plug into the regular Schwarzschild metric. Next, I take total derivative of LHS and RHS of first eqt'n and then solve for dr. This leaves me with a hopelessly complex expression for the metric that I have not been successful in reducing to the form given by the problem.

I can't find derivations of this anywhere on the internet (Wikipedia just states it). This makes me think that either the problem is exceedingly difficult, or exceedingly trivial. If it is trivial, I'm missing something. Any ideas?

 

[reverbtank]

I think the issue here is that I just need to face up to the messy algebra. Thanks.

 

[JosephKruskal]

I suggest that you look up the paper by Martin Kruskal in which he developed what you are looking for. The easiest way I know to look it up is to use Google Scholar.

It is very easy to do. First put "Martin Kruskal" in the entry place for "Authors" (use quote marks around the name). Even if there are other authors, you don't need to put them in.

Then put in what is necessary to narrow down the papers you get; Martin published a large number of papers. I think it would be enough to put in "Schwarzschild metric" and "Kruskal coordinates". With that you shouldn't get more than a few papers. Then get hold of that paper, and you should be able to read about it from the person who did it first.

 

[reverbtank]

Thanks for the post! It was a nice surprise, since I thought I had resolved this thread. I completed the assignment, but I will certainly look for the paper.

 

[stevebd1]

A little off topic but while the quantity of r is self explanatory in Kruskal coordinates, what quantity would be use for t? i.e. as the radius counts down from infinity to zero, how exactly is the quantity t expressed?

regards
Steve